Semilandmarks on Curves and Surfaces

Dean Adams, Iowa State University

15 October, 2019

Generalized Procrustes Analysis

\[\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\]

The Problem with Curves I

The Problem with Curves II

Mathematically-Defined Points: Problems

Mathematically-Defined Points: Problems

Mathematically-Defined Points: Problems

Example from: Gunz et al. (2005) in Modern morphometrics in physical anthropology.

Mathematically-Defined Points: Problems (Cont.)

Mathematically-Defined Points: Problems (Cont.)

Mathematically-Defined Points: Problems (Cont.)

Example from: Gunz et al. (2005) in Modern morphometrics in physical anthropology.

Semilandmarks and Procrustes Relaxation

Semilandmarks and Procrustes Relaxation

Semilandmarks and Procrustes Relaxation

Bookstein (1997) Med. Image Anal.

Procrustes Relaxation: Conceptual Procedure

Procrustes Relaxation: Conceptual Procedure

Procrustes Relaxation: Conceptual Procedure

Procrustes Relaxation: Conceptual Procedure

Procrustes Relaxation: Conceptual Procedure

Procrustes Relaxation: Conceptual Procedure

What Direction to Slide?

What Direction to Slide?

\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 \\ u_{2x} & 0 & 0\\ 0 & u_{3x} & 0 \\ 0 & 0 & u_{4x} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2y} & 0 & 0\\ 0 & u_{3y} & 0 \\ 0 & 0 & u_{4y} \\ 0 & 0 & 0 \\ \end{bmatrix}\]

What Direction to Slide?

\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 \\ u_{2x} & 0 & 0\\ 0 & u_{3x} & 0 \\ 0 & 0 & u_{4x} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2y} & 0 & 0\\ 0 & u_{3y} & 0 \\ 0 & 0 & u_{4y} \\ 0 & 0 & 0 \end{bmatrix}\]

How Far to Slide?

How Far to Slide?

where \(\small\mathbf{\mathcal{L}_p^{-1}}=\begin{bmatrix} \mathbf{L}_p^{-1} & 0 \\ 0 & \mathbf{L}_p^{-1} \end{bmatrix}\)

How Far to Slide?

where \(\small\mathbf{\mathcal{L}_p^{-1}}=\begin{bmatrix} \mathbf{L}_p^{-1} & 0 \\ 0 & \mathbf{L}_p^{-1} \end{bmatrix}\)

Note: \(\small\mathbf{Y^0}\) and \(\small\mathbf{Y_{ref}}\) are assembled by \(\small{x}\) then \(\small{y}\) coordinates
Statistical note: Sliding by \(\small{D}_{Proc}\) assumes independence of semilandmarks, as \(\small(\mathbf{U^T\mathcal{L}_p^{-1}U})^{-1}\mathcal{L}_p^{-1}=\mathbf{I}\) (this is not realistic).

Procrustes Relaxation: Flow of Computations

Procrustes Relaxation: Flow of Computations

Procrustes Relaxation: Flow of Computations

Procrustes Relaxation: Flow of Computations

Procrustes Relaxation: Flow of Computations

Semilandmarks: Differing Approaches

Mathematical comments: Using \(\small{BE}\) is akin to a weighted regression, which accounts for spatial proximity (non-independence) of adjacent landmarks. This is a more reasonable model, as one uses semilandmarks precisely to represent curves by an adjacent set of points which are explicitly not independent

Procrustes Relaxation: Refinements

Combining Points and Curves

Combining Points and Curves

\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & u_{2x} & 0 & 0& 0 \\ 0 & 0 & u_{3x} & 0 & 0 \\ 0 & 0 & 0 & u_{4x} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 0 & u_{2y} & 0 & 0 & 0\\ 0 & 0 & u_{3y} & 0 & 0 \\ 0 & 0 & 0 & u_{4y} & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\]

Note: with this notation, generalized inverses are required to complete GLS sliding (and process can be SLOW)
Note 2: geomorph uses very clever algebra to eliminate redundancies in computations to speed this process up considerably

Combining Landmarks and Curves: Example

Bookstein et al. (1999). Anatomical Record.

Semilandmark Example: The Data

Semilandmark Example: The Data

Semilandmark Example: Results

Semilandmarks on 3D Curves

\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 \\ u_{2x} & 0 & 0\\ 0 & u_{3x} & 0 \\ 0 & 0 & u_{4x} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2y} & 0 & 0\\ 0 & u_{3y} & 0 \\ 0 & 0 & u_{4y} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2z} & 0 & 0\\ 0 & u_{3z} & 0 \\ 0 & 0 & u_{4z} \\ 0 & 0 & 0 \end{bmatrix}\]

\[\tiny\mathbf{T}=(\mathbf{U^T\mathcal{L}_p^{-1}U})^{-1}\mathbf{\mathcal{L}_p^{-1}U^T}(\mathbf{Y^0-Y_{ref}})\] \[\tiny\mathbf{\mathcal{L}_p^{-1}}=\begin{bmatrix} \mathbf{L}_p^{-1} & 0 & 0 \\ 0 & \mathbf{L}_p^{-1} & 0 \\ 0 & 0 & \mathbf{L}_p^{-1} \end{bmatrix}\]

\[\tiny\mathbf{T}=\mathbf{U^T}(\mathbf{Y^0-Y_{ref}})\]

Semilandmarks on Surfaces

\[\tiny\mathbf{U}= \left[ \begin{array}{ccc|ccc} 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2x} & 0 & 0 & w_{2x} & 0 & 0 \\ 0 & u_{3x} & 0 & 0 & w_{3x} & 0 \\ 0 & 0 & u_{4x} & 0 & 0 & w_{4x} \\ 0 & 0 & 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 & 0 & 0\\ u_{2y} & 0 & 0 & w_{2y} & 0 & 0 \\ 0 & u_{3y} & 0 & 0 & w_{3y} & 0 \\ 0 & 0 & u_{4y} & 0 & 0 & w_{4y} \\ 0 & 0 & 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 & 0 & 0\\ u_{2z} & 0 & 0 & w_{2z} & 0 & 0\\ 0 & u_{3z} & 0 & 0 & w_{3z} & 0 \\ 0 & 0 & u_{4z} & 0 & 0 & w_{4z} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right]\]

NOTE: One can see that these matrices are becoming VERY large. That will lead to long computation time (though recall geomorph uses clever algorithms to speed this up! [ask Mike!])

Defining Semilandmarks on Surfaces

Defining Semilandmarks on Surfaces

Note: resampling methods required for significance testing as \(\small{p>>n}\)

Combining Points, Curves, and Surfaces

Complications: Shape Variables and Dimensionality

Complications: Shape Variables and Dimensionality

Note: #3 also required when # shape variables exceeds # specimens

Semilandmarks: Snake Example

Semilandmarks: Homo Example

Semilandmarks: Scallop Example

Semilandmarks: Scallop Example (Cont.)

Semilandmarks: Scallop Example (Cont.)

The Procrustes Paradigm